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<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>NE-cs.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13135 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:46 2006
The command was "/home/mccune/bin/prover9 -f cs.in NE.in".
]]></heading>

<proof number="1" length="25" max_count="19">

<comments><![CDATA[
% Proof 1 at 0.14 (+ 0.01) seconds: E.
% Length of proof is 25.
% Level of proof is 8.
% Maximum clause weight is 19.
% Given clauses 118.
]]></comments>

  <clause id="7">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      y = z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x * y != z * y
    ]]></literal>
    <literal><![CDATA[
      x = z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      A * B * C * B * A != B * A * C * A * B
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1)),demod(9(4))].">
      <j1 rule="para" parents="9 7"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(7,a,10,a)].">
      <j1 rule="hyper" parents="7 10"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x * y * (y @ z) != y * z
    ]]></literal>
    <literal><![CDATA[
      z = x
    ]]></literal>
    <justification jstring="[para(10(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="10 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,1)),demod(9(2),9(5)),flip(a)].">
      <j1 rule="para" parents="10 9"/>
      <j2 rule="demod" parents="9 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(15(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="15 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(15(a,1),9(a,1,1)),flip(a)].">
      <j1 rule="para" parents="15 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      x * (x @ (y @ z)) = x
    ]]></literal>
    <justification jstring="[hyper(16,a,11,a)].">
      <j1 rule="hyper" parents="16 11"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      x @ (y @ z) = x @ x
    ]]></literal>
    <justification jstring="[hyper(33,a,70,a),flip(a)].">
      <j1 rule="hyper" parents="33 70"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x @ (y @ z)) * u = u * (x @ x)
    ]]></literal>
    <justification jstring="[para(81(a,2),11(a,1,1))].">
      <j1 rule="para" parents="81 11"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      y @ y = x
    ]]></literal>
    <justification jstring="[para(15(a,1),17(a,2)),demod(70(3))].">
      <j1 rule="para" parents="15 17"/>
      <j2 rule="demod" parents="70"/>
    </justification>
  </clause>

  <clause id="113">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(13,a,36,a)].">
      <j1 rule="hyper" parents="13 36"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(105,a,113,a)].">
      <j1 rule="hyper" parents="105 113"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      (x @ (y @ z)) * u = u
    ]]></literal>
    <justification jstring="[para(81(a,2),113(a,1,1))].">
      <j1 rule="para" parents="81 113"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(122)].">
      <j1 rule="new_symbol" parents="122"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_demod(98),demod(130(3),134(1)),flip(a)].">
      <j1 rule="back_demod" parents="98"/>
      <j2 rule="demod" parents="130 134"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="150">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[back_demod(81),demod(134(3))].">
      <j1 rule="back_demod" parents="81"/>
      <j2 rule="demod" parents="134"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x * y * z * (y @ x) = y * x * z
    ]]></literal>
    <justification jstring="[para(10(a,1),19(a,1,2,2)),demod(150(6),136(6))].">
      <j1 rule="para" parents="10 19"/>
      <j2 rule="demod" parents="150 136"/>
    </justification>
  </clause>

  <clause id="814">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(160(a,2),12(a,1)),demod(9(11),9(10),10(10)),xx(a)].">
      <j1 rule="para" parents="160 12"/>
      <j2 rule="demod" parents="9 9 10"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
